11 Central Projections Central projections are similar to parallel projections in that they also associate points in one set with points in another set by projection lines that pass through the associated points, with the difference that the projection lines all pass through a given point. Central projections, behave differently too, in ways that make them, at the same time, difficult to construct manually, and interesting as visual objects. 11.1 BASIC PROPERTIES 11.1.1 Central projection of a line on a line The set of all lines that pass through any point C is a pencil of lines through C. For any two lines coplanar with a pencil of lines through a point C on neither line, for any point on one line there is a line in the pencil maps it to a point on the other. This is called a central projection and C is the center of projection. P' 11-1 Illustrating a central projection P pencil of lines C center of projection As is the case with parallel projections, the lines may or may not be parallel, and these distinctions produce important differences in the resulting mappings. Case: Parallel lines Consider the case where the lines are parallel (see Figure 11-2). Q' Q' P' P' C Q P P P'Q' = k PQ, where k is a constant Q C 11-2 Central projections between parallel lines As C is not on either line, every point P on one line defines, together with C, a line that passes through these two points; it is therefore not parallel to either line; moreover, it passes through a point on the other line, which is the unique image of P under the projection. By the same argument, every point on the second line is the image of a unique point on the first. That is, the projection is maps the points one on one. Using similar triangles, we can show that a central projection of a line on a parallel line multiplies distances by a constant factor, k. This factor is not normally 1, except in the trivial case where the lines coincide or when C is equidistant to both lines. We summarize these observations in the following assertion: Property 11-1 A central projection of a line on a parallel line uniquely maps between the points on the lines and multiplies distances by a constant factor (generally not 1). It follows, as an immediate consequence, that this type of projection preserves between- ness, and therefore, maps line segments onto line segments, rays on rays and lines onto lines. These are the same properties that are also preserved by a parallel projection between lines (which do not have to be parallel); a difference between the two types of projections is that a parallel projection between concurrent lines fixes a point (the point of intersection), while a central projection between parallel lines, of course, cannot fix a point. 310 Case: Concurrent lines The case becomes more intricate when we consider the central projection of a line on a concurrent line. See Figure 11-3. l' l P vanishing point on l C V m' Q P' V' Q' m vanishing point on m (i) Vanishing points l R Q C P' R' P Q' m (ii) Between-ness is not preserved 11-3 Central projection of a line onto a concurrent line Let l and m be two concurrent lines and C, the center of projection. Consider the point of intersection, V, where line, m', parallel to m, passing through C, meets l (see Figure 11-3i). m' does not intersect m at any point; consequently, V has no image under this projection. All other points of l have a unique image on m. If such a point P on l moves closer and closer to V, its image P' moves farther and farther away on m. For this reason, V is called a vanishing point on l. Likewise, the point, V', where the line l', parallel to l, passing through C, meets m cannot be the image of any point on l. All other points of m are the unique image of a point on l. V' is called the vanishing point on m. 311 Figure 11-3ii illustrates the fact that a central projection of a line on a concurrent line does not preserve betweenness: for example, point Q is between P and R on l, but its image, Q’, is not between the images of the other two points, P' and R', on m. An aside: cross ratio One of the more important properties of a central projection of a line is the following important invariant: preserving ‘cross ratio’ of distances. Let P, Q, R and S be four points on l with P and Q distinct. Then, the cross ratio of the four points is defined as the ratio: (P, Q, R, S) = (PR × QS) / (PS × QR), where lengths of the segments are measured along the same direction. P' Q' R' S' S'P' R'Q' = 1.89 = 0.37 S' S'Q' R'P' R'Q' S'P' R' ⋅ = 0.70 Q' R'P' S'Q' P' RQ SP S'P' R'Q' = 0.48 = 1.45 = 1.45 = 0.48 S RP SQ S'Q' R'P' Q R P RQ SP R'Q' S'P' ⋅ = 0.70 ⋅ = 0.70 RP SQ R'P' S'Q' C R'Q' S'P' = 0.39 = 1.81 S' R' Q' P' R'P' S'Q' R'Q' S'P' ⋅ = 0.70 R'P' S'Q' 11-4 Illustrating the cross-ratio invariance If P', Q', R' and S' are the images of the four points on any line m under a central projection of l, then (P', Q', R', S') = (P, Q, R, S). The cross ratio plays an important part in the development of projective geometry; however, we will not consider this any further in this course. 312 11.1.2 Central projection of a plane on a plane Just as the case of consider a projection of a line on a line through a central projection, we can consider the central projection of a plane onto a plane by mapping every point on one plane to unique point on another by a pencil lines that pass through a point C not on either plane. Again, C is called the center of the projection. Case: Parallel planes Let us again first consider the case when the two planes are parallel (see Figure 11-5). Consider a line l in p and a point P on l. The line in the pencil of the projection that passes through P defines together with l a plane to which all of the lines in the pencil passing through a point on l belong. This plane intersects p' at a line, which is the image of l under the projection. A central projection of a plane on a parallel plane thus maps lines on lines. C P l P' l' 11-5 Central projection of a plane on a parallel plane We can observe furthermore that this projection uniquely maps between the points in both planes. Therefore, the images of two parallel lines cannot intersect, while the images of two concurrent lines must be concurrent. The projection thus preserves parallelism and concurrence between lines. Consider a line l and its image, l’. As they are on parallel planes, they must be parallel. That is, they are coplanar (because they belong both to the plane defined by them), and their points are associated by a central projection between parallel and coplanar lines. By Property 11-1, the projection multiplies distances by a constant factor. We can show that this factor is constant for the entire projection. We again summarize the findings as follows: 313 Property 11-2 A central projection of a plane on a parallel plane uniquely maps between the points in the planes. It maps lines on lines and multiplies distances by a constant factor. A consequence is that a central projection between parallel planes preserves between- ness between points, and parallelism and concurrence for lines. It also maps conic sections onto conic sections of the same type. Case: Concurrent planes The situation is more intricate for a central projection of a plane on a concurrent plane (see Figure 11-6). v vanishing lines C line parallel to the v' vanishing line contains no P vanishing points and likewise its image P' line not parallel to the vanishing line contains a vanishing point and likewise its image 11-6 Central projection of a plane on a concurrent plane The plane through C parallel to p' intersects p at a line, v. No point on v can have an image under the projection because the lines in the pencil through these points are parallel to p'. v is consequently called a vanishing line and every point on it a vanishing point. Likewise, the plane through C parallel to p intersects p' at a line, v', so that no point on v’ is the image of a point on p. v' and the points on it are again called vanishing line and vanishing points, respectively. Every point on p that is not on v has a unique image on p', and every point on p’ that is not on v’ is the image of a unique point on p. The projection thus establishes a unique correspondence or mapping between the points on p and p' that are not on v or v'. Consider a line l on p that is not the vanishing line.